There are a bunch of different notions of length/depth in ring theory: Projective length, Artinian length, local depth, etc. If we take length to mean Artinian length, then Charles is right: The Artinian length of a finite-dimensional commutative algebra is just its dimension. Every such algebra is a direct sum of local ones, and you can chip away at each local summand of the ring from the bottom end, one dimension at a time.
The local algebras that have a description that looks as nice as $\mathbb{F}[x]/(x^n)$ are the toric ones. These local algebras are $\mathbb{F}[\vec{x}]$ divided by an ideal generated by monomials and a basis of monomials. You can make a diagram of the exponents of the monomials that aren't killed by the ideal. If the ring has $n$ generators, then the diagram is a stable stack of blocks in the $n$-dimensional orthant. For example, the algebra $\mathbb{F}[x,y]/(x^3,x^2y,y^3)$ has a basis of seven monomials: 1, $x$, $x^2$, $y$, $xy$, $y^2$, $xy^2$. The diagram of these monomials looks like this:
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I have put a 1 at the corner in the diagram corresponding to the monomial 1.
What is easy to forget is that all finite-dimensional local algebras with one generator in $m/m^2$ are of this form, but with more variables these are just special examples.
I was looking at the second part of the question first, interesting higher-dimensional examples. Here is a list of inequivalent examples of local rings $R$ (not necessarily algebras) that have length 2 or 3 and such that $R/m = \mathbb{Z}/p$:
Length 2:
- $(\mathbb{Z}/p)[x]/(x^2)$
- $\mathbb{Z}/p^2$
Length 3:
- $(\mathbb{Z}/p)[x]/(x^3)$
- $(\mathbb{Z}/p)[x,y]/(x^2,xy,y^2)$
- $(\mathbb{Z}/p^2)[x]/(px,x^2)$
$(\mathbb{Z}/p^2)[\sqrt{p}]$$\mathbb{Z}[\sqrt{p}]/p^{3/2}$- $\mathbb{Z}/p^3$
(Edit: My notation for #4 was not strictly correct.)
I think, although I can't really speak with authority, that these are all of them. The point is that among algebras over $\mathbb{Z}/p$, the length is too small to see anything non-toric. But you can also have local rings that look like these toric local algebras (which I listed first), but have carries. The most creative one is the fourth one of length 3, namely $(\mathbb{Z}/p^2)[\sqrt{p}]$. You can express an element of this ring as three digits in base $p$, say $d_2d_1d_0$. Then addition carries from $d_0$ to $d_2$.
I would also guess that all of these generalize to $\mathbb{F}_q$, using the Witt vector construction in the cases with carries. And maybe it is again all of them.